a comparison of modern hyperbolic methods for semiconductor … · 2019. 8. 1. · hyperbolic...

A Comparison of Modern Hyperbolic Methods forSemiconductor Device Simulation: NTK Central Scheme Vs.

CLAWPACK

CARL L. GARDNER*, ANNE GELB† and JUSTIN HERNANDEZ

Department of Mathematics, Arizona State University, Tempe AZ 85287-1804, USA

(Received 1 May 2001; Revised 1 April 2002)

Two modern hyperbolic methods—a second-order Godunov method in the software packageCLAWPACK and the second-order Nessyahu–Tadmor–Kurganov (NTK) central scheme—arecompared for simulating an electron shock wave in the classical hydrodynamic model forsemiconductor devices.

The NTK central scheme, which does not employ Riemann problem solutions, is described in detail.Special attention is paid in both methods to handling the source terms in the hydrodynamic model.CLAWPACK incorporates the source terms by a splitting method, while our version of the NTK schemeis unsplit.

Keywords: Numerical methods for hyperbolic PDEs; CLAWPACK; Hyperbolic central schemes; NTKcentral scheme

INTRODUCTION

The classical hydrodynamic model has become a standard

industrial simulation tool which incorporates important

“hot electron” phenomena in submicron semiconductor

devices. The hydrodynamic model consists of non-linear

hyperbolic conservation laws for particle number,

momentum, and energy (with a heat conduction term),

coupled with Poisson’s equation for the electrostatic

potential. The non-linear hyperbolic modes support shock

waves—“velocity overshoot” in the parlance of semi-

conductor device physicists.

To accurately compute solutions including sharp

resolution of shock waves, we implemented the hydro-

dynamic model equations in LeVeque’s software package

CLAWPACK‡ (Conservation LAWs PACKage) and in our

general gas dynamical code which uses the Nessyahu–

Tadmor–Kurganov (NTK) central scheme [1]. CLAW-

PACK consists of routines for solving time-dependent non-

linear hyperbolic conservation laws based on higher order

Godunov methods and (approximate) Riemann problem

solutions, while the NTK scheme solves conservation laws

using a modified Lax–Friedrichs central difference

method without appealing to Riemann problem solutions.

We will compare hydrodynamic model steady-state

simulations (including shock waves) using CLAWPACK

and the NTK central scheme of the nþ –n–nþ diode,

which models the channel of a field effect transistor. Steady

state is achieved by simulating in time to equilibrium.

A splitting method is used to incorporate the source

terms in the hydrodynamic model for CLAWPACK, while

our version of the NTK scheme is unsplit. In both numerical

codes, we use an implicit method for the source terms.

THE CLASSICAL HYDRODYNAMIC MODEL

The hydrodynamic model treats the propagation of

electrons in a semiconductor device as the flow of a

charged compressible fluid. The model plays an important

ISSN 1065-514X print/ISSN 1563-5171 online q 2002 Taylor & Francis Ltd

DOI: 10.1080/1065514021000012336

*Corresponding author. Tel.: þ1-480-965-0226. Fax: þ1-480-965-0461E-mail: [email protected]. Research supported in part by the NationalScience Foundation under grant DMS-9706792.

†E-mail: [email protected]. Research supported in part by the Sloan Foundation and in part by the National Science Foundation under grantDMS-0107428.

‡http://www.amath.washington.edu/ , claw/. R.J. LeVeque’s CLAWPACK website.

VLSI Design, 2002 Vol. 15 (4), pp. 721–728

role in predicting the behavior of electron flow in semi-

conductor devices since it exhibits hot carrier

effects missing in the standard drift-diffusion model.

The hydrodynamic description should be valid for devices

with active regions * 0.05 microns.

The hydrodynamic model is equivalent to the equations

of electro-gas dynamics:

›

›tðmnÞ þ

›

›xi

ðmnuiÞ ¼ 0 ð1Þ

›

›tðmnujÞ þ

›

›xi

ðmnuiujÞ þ›P

›xj

¼ 2n›V

›xj

2mnuj

tp

ð2Þ

›W

›tþ

›

›xi

ðuiðW þ PÞ þ qiÞ

¼ 2nui

›V

›xi

2W 2 3

2nT0

� �tw

ð3Þ

7·ðe7VÞ ¼ e2ðN 2 nÞ ð4Þ

where m is the effective electron mass, n is the electron

density, ui is the velocity, mnui is the momentum density,

P ¼ nT is the pressure (Boltzmann’s constant kB is set

equal to 1), T is the temperature, V ¼ 2ef is the potential

energy, f is the electrostatic potential, e . 0 is the

electronic charge,

W ¼3

2nT þ

1

2mnu2 ð5Þ

is the energy density, qi is the heat flux, and T0 is

the temperature of the semiconductor lattice. Indices i, j

equal 1, 2, 3, and repeated indices are summed over.

Equation (1) expresses conservation of electron number

(or mass), Eq. (2) expresses conservation of momentum,

Eq. (3) expresses conservation of energy, and Eq. (4) is

Poisson’s equation. The last terms in Eqs. (2) and (3)

represent electron scattering, which is modeled by the

standard relaxation time approximation, with momentum

and energy relaxation times tp and tw.

The non-linear conservations laws Eqs. (1)–(3) are

simply the Euler equations of gas dynamics for a charged

gas with polytropic gas constant g ¼ 5=3 in a electric field,

with the addition of heat conduction and relaxation terms.

In this investigation we take tp and tw to be constant; we

also set the heat flux qi ¼ 0 since we are interested here in

numerical methods which sharply resolve shock waves.

Note that the hydrodynamic model equations have source

terms on the right-hand sides of Eqs. (2) and (3) from the

relaxation time terms and from the coupling to the electric

field E ¼ 27f:The electron gas has a soundspeed and the electron flow

may be either subsonic or supersonic. In general a shock

wave develops at the transition from supersonic flow to

subsonic flow.

THE n1 –n–n1 DIODE

The hydrodynamic model has been extensively used to

study the nþ –n–nþ diode which models the channel

of a field effect transistor. The diode begins with a heavily

doped n þ source region, followed by a lightly doped n

channel region, and ends with an n þ drain region.

Hydrodynamic model simulations of a steady-state

electron shock wave [2] in a one micron Si semiconductor

device at 77 K were validated in Ref. [3] by a Monte Carlo

simulation of the Boltzmann equation using the DAMO-

CLES program [4]. A shock profile develops in the

channel as the supersonic flow on entering the channel

breaks to a subsonic flow, just like gas dynamical flow in a

Laval nozzle. The electron shock wave is analogous to

the gas dynamical shock wave in the Laval nozzle, where

the nþ –n–nþ doping of the diode corresponds to the

converging/diverging geometry of the Laval nozzle.

The steady-state upwind shock simulations presented in

Ref. [2] were reproduced in Ref. [5] using a time-

dependent “essentially non-oscillatory” (ENO) upwind

scheme, a higher order Godunov method.

The shock computations imply that the electron shock

waves are an integral part of the hydrodynamic model.

The shock waves allow higher electron velocities to

develop in the channel, and provide a richer space charge

structure in the device.

For the transonic computations presented below, we

take a GaAs diode at T0 ¼ 300 K with a 0.25 micron

source, a 0.25 micron channel, and a 0.25 micron drain. In

the n þ region, the doping density N ¼ 5 £ 1017 cm23;while in the n region N ¼ 2 £ 1015 cm23: In GaAs, the

effective electron mass m ¼ 0:063 me at 300 K, where me

is the electron mass, and the dielectric constant e ¼ 12:9:We set tp ¼ tw ¼ 0:2 picoseconds. Since the flow is

subsonic at inflow and outflow, we impose the boundary

conditions n ¼ N at the left and right, T ¼ T0 at the left,

and f ¼ 0 at the left and f ¼ 1 volt at the right.

CLAWPACK

Our main interest here lies with the non-linear hyperbolic

modes of the hydrodynamic model, since they support

shock waves. Implementing a higher order Godunov

method via CLAWPACK yields sharp resolution of these

discontinuities.

CLAWPACK is second-order accurate in regions of

smooth flow and first-order accurate near discontinuities

for the homogeneous conservation laws of gas dynamics.

The best results were obtained using the Roe approximate

Riemann solver and the van Leer flux limiter.

The hydrodynamic model has inhomogeneous source

terms and we use a splitting method to take them into

account. The one-dimensional hydrodynamic model

conservation laws take the form

qt þ f ðqÞx ¼ c ðqÞ ð6Þ

C.L. GARDNER et al.722

where the conserved quantities q ¼ ðmn;mnu;WÞ; the flux

f ðqÞ ¼ ðmnu;mnu2 þ P; uðW þ PÞ ð7Þ

and the source terms

cðqÞ ¼ 2 0; enE þmnu

tp

; enE þW 2 3

2nT0

tW

� �: ð8Þ

LeVeque [6] suggests using a standard splitting or

fractional step method in which one alternates between

solving the homogeneous equation

qt þ f ðqÞx ¼ 0 ð9Þ

and the ordinary differential equation (ODE)

qt ¼ c ðqÞ: ð10Þ

We use the second-order accurate implicit trapezoidal

rule method for solving the ODE [Eq. (10)] to insure

accuracy and stability. Non-linear source terms are

handled by Newton iteration.

Since the hydrodynamic model equations are coupled

with Poisson’s equation for the electrostatic potential, a

elliptic solve must also be performed. To discretize

Poisson’s equation (4) we use a second-order accurate

central difference for the approximation of the second

derivative Vxx. The potential is then updated by means of a

standard direct tridiagonal solver.

Solving the hydrodynamic model Eqs. (1)–(4) over one

timestep Dt involves the following steps:

1. Solve the source term ODE [Eq. (10)] over a timestep

Dt/2.

2. Using step 1 as initial data, solve the gas dynamical

homogeneous conservation laws [Eq. (9)] over Dt.

3. Solve Poisson’s equation (4), using the results from

step 2 as initial data.

4. Solve the source term ODE [Eq. (10)] over a timestep

Dt/2, using the previous results in steps 2 and 3 as

initial data.

The Strang splitting yields second-order accurate

results; in practice steps 1 and 4 are combined except in

the first and last timesteps.

Using CLAWPACK, we performed simulations of the

nþ –n–nþ diode with 300 Dx. The CLAWPACK results

are compared in the Figures with the results of the NTK

central scheme with 300 Dx.

Figures 1–5 present the simulation of the electron shock

wave. The shock wave develops at x < 0:2 microns in

Fig. 1 and with CLAWPACK is spread out over

approximately one or two Dx (see Fig. 2).

Figure 3 displays the doping profile N and steady-state

solution for electron density n. Because of the high density

contrast of the background doping, the shock wave in

density is downplayed. Note the slight cooling in Fig. 4 as

the electrons overcome a small potential barrier at the first

junction, and the dramatic increase in temperature as the

electrons heat up in the channel before entering the drain.

Figure 5 depicts the effects on the electric field of the

electron shock wave and of the space charge n 2 N at

the junctions.

NTK CENTRAL SCHEME

High resolution non-oscillatory central schemes have

recently become popular in solving non-linear hyperbolic

conservations laws. We first consider the one-dimensional

scalar conservation law

ut þ f ðuÞx ¼ 0: ð11Þ

The computation is performed over cells of width

Dx ¼ xjþ1=2 2 xj21=2 given by the piecewise constant

FIGURE 1 Electron velocity in 107 cm/s using CLAWPACK (solid) vs. the NTK central scheme (dotted solution values). x is in 0.1 microns for allfigures.

NTK CENTRAL SCHEME VS. CLAWPACK 723

approximate solution

�uðx; tÞ ¼ ujðtÞ; xj212# x # xjþ1

2ð12Þ

where gridpoints are labeled by j ¼ 0; 1; . . .; N. The

original Nessyahu–Tadmor (NT) scheme [7] was derived

from the prototypical staggered version of the Lax–

Friedrichs (LxF) scheme,

ujþ12ðt þ DtÞ ¼

1

2ðujðtÞ þ ujþ1ðtÞÞ2 l½ f ðujþ1ðtÞÞ

2 f ðujðtÞÞ� ð13Þ

where l ¼ Dt=Dx: As noted in Ref. [7], central schemes

integrate over the entire Riemann fan, taking into account

both left and right traveling waves. Hence there is no need

to employ a Riemann solver. By exploiting this freedom,

we show how high resolution central schemes can be

effectively adapted to numerically solve conservation

laws with source terms. Specifically, since no Riemann

solvers need to be employed for the homogeneous

hyperbolic part of the system, the source terms can be

handled implicitly in an unsplit way, thereby reducing the

overall computational time. This technique will be

demonstrated in the application of a high resolution

central scheme to the hydrodynamic model for semi-

conductor devices. If the source terms are non-linear, then

Newton iteration can be performed, which now involves

repeating the gas dynamical step as well. This repetition

will be computationally expensive, so a semi-implicit

method for the source terms may be substituted.

The idea behind the NT scheme is to reduce the amount

of inherent dissipation in the LxF scheme [Eq. (13)].

This is accomplished by replacing the piecewise constant-

solutions in Eq. (12) with piecewise linear approximations

Ljðx; tÞ ¼ ujðtÞ þ ðx 2 xjÞu0

j

Dx; xj21

2# x # xjþ1

2ð14Þ

FIGURE 2 Closeup of the electron shock wave in velocity.

FIGURE 3 Electron density using CLAWPACK (solid) vs. the NTK central scheme (dotted) and doping profile in 1018 cm23.


where the numerical derivative at xj is u0j=Dx: A second-

order scheme is derived by evolving this linear interpolant

in time to produce


1

2ðujðtÞ þ ujþ1ðtÞÞ þ

1

8ðu0

j 2 u0jþ1Þ

2 l f ujþ1 t þDt

2

� �� 2 f uj t þ

Dt

2

� �� :

ð15Þ

The CFL condition is satisfied by

lx

max›f

›u

�� , 1

2: ð16Þ

Analogously for systems we have

lx

maxr ðAðuðx; tÞÞÞ ,1

2ð17Þ

where r(·) denotes the spectral radius and A(u ) is the

Jacobian ›f=›u of the flux.

By Taylor expansion, the midpoint values can be

approximated by

uj t þDt

2

� �¼ uj 2

1

2lf 0j ð18Þ

where the numerical flux derivative at xj is f 0j=Dx: Second-

order accuracy is ensured if the numerical derivative and

numerical flux derivative satisfy

u0j

Dx¼ uxjx¼xj

þ OðDxÞ ð19Þ

f 0j

Dx¼ f ðuÞxju¼uðxjÞ þ OðDxÞ: ð20Þ

Guidelines are provided in Ref. [7] for determining the

numerical derivatives [Eqs. (19) and (20)] to ensure that

FIGURE 5 Electric field in volts/cm using CLAWPACK (solid) vs. the NTK central scheme (dotted).

FIGURE 4 Electron temperature in electron volts using CLAWPACK (solid) vs. the NTK central scheme (dotted).


Eq. (15) is TVD (total variation diminishing) in the scalar

case. In practice these conditions are met by

u0j ¼ minmod uDuj21

2;

1

2ðujþ1 2 uj21Þ; uDujþ1

2

� ð21Þ

f 0j ¼ minmod uDf j212;

1

2ðf jþ1 2 f j21Þ; uDf jþ1

2

� ð22Þ

where u [ ½1; 2� and Dujþ1=2 ¼ ujþ1 2 uj: One can

instead use the standard flux derivatives f 0j ¼ AðujÞu0j:

The NT scheme is easily implemented as a predictor–

corrector method

uj t þDt

2

� �¼ ujðtÞ2

l

2f 0j ð23Þ


1

2ðujðtÞ þ ujþ1ðtÞÞ þ

1

8ðu0

j 2 u0jþ1Þ

2 l f ujþ1 t þDt

2

� �� 2 f uj t þ

Dt

2

� �� :

ð24Þ

The amount of dissipation produced in the NT and NTK

central schemes is inversely proportional to u.

Because of its simplicity in construction and ability to

produce second-order accuracy in smooth regions without

requiring explicit knowledge of the characteristic structure

of the equations, the NT scheme has recently become a

popular choice for simulating semiconductor devices

using the hydrodynamic model [8–10]. Here we modify

the NT scheme for the hydrodynamic model in two

important ways.

First, we adopt the semi-discrete formulation of the NT

scheme introduced in Ref. [1], which we will refer to as

the NTK scheme. In this formulation, the staggered grid

occurs between timesteps. Hence all computation is

performed only on the non-staggered gridpoints, eliminat-

ing the more complicated staggered grid construction used

in all previous applications to semiconductor devices.

Another significant advantage is that tighter control

volumes are used at shock formations, producing sharper

resolution in non-smooth regions. The detailed derivation

of this method can be found in Ref. [1] and is briefly

summarized below. We note that although different orders

of the Runge–Kutta method were applied there to the

semi-discrete NTK method, we improve on both the

efficiency and complexity by implementing the second-

order Adams – Bashforth scheme. No reduction of

accuracy is visible.

Second, in all previous applications of the NT method

to the hydrodynamic model, a splitting technique was used

to perform temporal integration. The source term step was

computed first, and then the advection step was computed

using the results from the source term step as initial

conditions. This is also the usual approach taken by the

Godunov methods for the hydrodynamic model. For

steady-state problems, splitting causes the solution to vary

slightly because of the additional partial timestep required

to complete the method (highly accurate steady-state

solutions can be obtained by modifying the method to

avoid splitting as steady state is approached—see, e.g.

Ref. [11]). Furthermore, two partial steps are required at

each time level, increasing the work and computational

time. By taking advantage of the Riemann solver free

approach in the NT and NTK methods, we are able to

avoid splitting.

We will briefly review the NTK semi-discrete

formulation. Again we evolve in time the piecewise linear

solution [Eq. (14)] at time level n based on cell averages unj

and the reconstructed approximate derivatives ðuxÞnj : We

define

uþjþ1

2

¼ ujþ1 2Dx

2ðuxÞjþ1 ð25Þ

u2jþ1

2¼ uj þ

Dx

2ðuxÞj ð26Þ

as the corresponding left and right intermediate cell

values. If C defines a curve in phase space connecting

u2jþ1=2 and uþ

jþ1=2via the Riemann fan, then the maximum

local speed of propagation at the cell boundary xjþ1=2 is

given by

ajþ12¼

u[Cmaxr

›f

›u

� �: ð27Þ

Knowledge of the maximum local speed allows us to

obtain tighter control volumes at shock formations. The

semi-discrete NTK central scheme for the conservation

law [Eq. (11)] can be written in conservation form as

d

dtujðtÞ ¼ 2

Hjþ122 Hj21

2

Dxð28Þ

with numerical flux

Hjþ12¼

f uþjþ1

2

ðtÞ �

þ f u2jþ1

2

ðtÞ �

2

2ajþ1

2ðtÞ

2uþ

jþ12

ðtÞ2 u2jþ1

2ðtÞ

�: ð29Þ

The maximum local speed is typically taken to be

ajþ12¼ max r

›f

›uðujðtÞÞ

� �; r

›f

›uðujþ1ðtÞÞ

� �� : ð30Þ

As in the case of the original NT scheme, the NTK

scheme is TVD in the scalar case if the approximate

derivatives ðuxÞjðtÞ ¼ u0j=Dx satisfy Eq. (21). Additionally,

the intermediate flux value f ðuþjþ1=2

ðtÞÞ can either be

computed as the traditional flux value of uþjþ1=2

ðtÞ or can be

approximated by fþjþ1=2 ¼ f jþ1 2 Dx=2ðf xÞjþ1; with ðf xÞjþ1

computed as in Eq. (22).


We write the hydrodynamic model conservation laws as

qt þ f ðqÞx ¼ Gq ð31Þ

where

G ¼ 2

0 0 0

Em

1tp

0

32

T0

mtw

Em

1tw

0BBB@

1CCCA: ð32Þ

As noted above, one of the main restrictions in using

CLAWPACK is that the hydrodynamic model conserva-

tion laws are solved by splitting the system into advection

and source term steps. This is necessary because the

Riemann problems that must be solved at each timestep

for the homogeneous part of the system cannot be

coupled with the source terms. Until now, the NT method

was also incorporated with a source term step. However,

this splitting is completely unnecessary since the NT and

NTK schemes do not require Riemann solvers. In fact,

the relatively simple structure of the source terms in the

hydrodynamic model enables us to use a mixed explicit–

implicit method, which both improves the stability

restriction on the timestep Dt and decreases the total

number of timesteps. Furthermore, we rid our steady-

state solution of the undesirable side effects of splitting.

Our method works as follows:

1. The semi-discrete formulation can be written as

qt ¼ C½q� þ Gq ð33Þ

where C[q ] denotes the spatial discretization of

2 f(q )x and is computed at the nth time level using

Eq. (28) as

C½qnj � ¼ 2

Hnjþ1

2

2 Hnj21

2

Dx: ð34Þ

2. Poisson’s equation (4) is solved using a standard

tridiagonal matrix solver to obtain Vnj at the nth time

level for each j ¼ 1; . . .; N 2 1. The electric field is

computed as the centered difference approximation

Enj ¼ ðVn

jþ1 2 Vnj21Þ=ð2DxÞ and used in Eq. (32).

3. Time integration is performed by applying the

second-order Adams–Bashforth method for the

conservation part and the trapezoidal rule for

the source terms as

qnþ1j ¼ qn

j þ1

2ð3C½qn

j �2 C½qn21j �Þ þ

1

2ðGqnþ1

j

þ Gqnj Þ: ð35Þ

Figures 1–5 show the profiles of the steady-state

solution. We chose the limiting parameter in Eqs. (21) and

(22) to be u ¼ 1:2:We note that the choice for the limiting value u is not

arbitrary. Figure 6 illustrates the effects of different values

of u on the velocity profile of the electron shock wave. The

simulations of two-dimensional Riemann problems in Ref.

[12] using the NTK central scheme indicate that the best

range of u for two-dimensional gas dynamics is 1 # u #

1:3: The optimal value for resolving the electron shock

wave is u ¼ 1:2: For u $ 1:25 there is an overshoot in the

shock wave, while for u , 1:2 the discontinuity is less

sharp.

CONCLUSION

CLAWPACK resolves the electron shock wave over

approximately one or two Dx (see Fig. 2), while the NTK

central scheme requires three or four Dx, which is typical

of shock resolution for the central schemes. It should be

noted however that the shock simulation with the NTK

FIGURE 6 Closeup of the shock wave in velocity using the NTK scheme for (from bottom to top at x ¼ 1:95) u ¼ 1, 1.2, 1.25, and 1.3.


scheme exhibits no spurious oscillations or overshoot with

u ¼ 1:2:On the one hand, the central schemes are easier to

program than the (higher-order) Godunov methods, are

typically faster by a factor of two or more, and can handle

source terms without splitting. Further, even minor

modifications of the physics of a model can lead to

major changes in the eigenstructure of the partial

differential equation (PDE) system and in the implemen-

tation of higher-order Godunov methods and their

Riemann solvers, while the central schemes remain

unchanged and are in this sense “universal”. On the other

hand, it is easy to incorporate source terms in CLAW-

PACK, and once a system of PDEs is working in

CLAWPACK in one spatial dimension, it is transparent to

extend the simulations to two and three dimensions, and to

make use of automatic mesh refinement.

The current vs. voltage curves—which are the main

focus of semiconductor device simulation in the

microelectronics industry—predicted by CLAWPACK

and the NTK scheme agree to better than six parts in ten

thousand, so unless for mathematical reasons one needs to

resolve shock waves over 1–2 Dx, either method may be

used to advantage.

Acknowledgements

We would like to thank Eitan Tadmor for valuable

discussions.

References

[1] Kurganov, A. and Tadmor, E. (2000) “New high-resolution centralschemes for nonlinear conservation laws and convection-diffusionequations”, Journal of Computational Physics 160, 214–282.

[2] Gardner, C.L. (1991) “Numerical simulation of a steady-stateelectron shock wave in a submicrometer semiconductor device”,IEEE Transactions on Electron Devices 38, 392–398.

[3] Gardner, C.L. (1993) “Hydrodynamic and Monte Carlo simulationof an electron shock wave in a one micrometer nþ –n–nþ diode”,IEEE Transactions on Electron Devices 40, 455–457.

[4] Fischetti, M.V. and Laux, S.E. (1988) “Monte Carlo analysis ofelectron transport in small semiconductor devices including band-structure and space-charge effects”, Physical Review B 38,9721–9745.

[5] Fatemi, E., Gardner, C.L., Jerome, J.W., Osher, S. and Rose, D.J.(1991) “Simulation of a steady-state electron shock wave in a

submicron semiconductor device using high-order upwindmethods”, Computational Electronics: Semiconductor Transportand Device Simulation (Kluwer Academic Publishers, Boston), pp27–32.

[6] LeVeque, R.J. (1992) Numerical Methods for Conservation Laws(Birkhauser Verlag, Basel).

[7] Nessyahu, H. and Tadmor, E. (1990) “Non-oscillatory centraldifferencing for hyperbolic conservation laws”, Journal ofComputational Physics 87, 408–463.

[8] Anile, A.M., Nikiforakis, N. and Pidatella, R.M. (2000) “Assess-ment of a high resolution centered scheme for the solution ofhydrodynamical semiconductor equations”, SIAM Journal onScientific Computing 22, 1533–1548.

[9] Anile, A.M., Romano, V. and Russo, G. (2000) “Extendedhydrodynamical model of carrier transport in semiconductors”,SIAM Journal on Applied Mathematics 61, 74–101.

[10] Romano, V. and Russo, G. (2000) “Numerical solution forhydrodynamical models of semiconductors”, Mathematical Modelsand Methods in Applied Sciences 10, 1099–1120.

[11] LeVeque, R.J. and Pelanti, M. (2001) “A class of approximateRiemann solvers and their relation to relaxation schemes”, Journalof Computational Physics 172, 572–591.

[12] Kurganov, A. and Tadmor, E. “Solution of two-dimensionalRiemann problems for gas dynamics without Riemann problemsolvers.” To appear.

Carl L. Gardner is Professor of Mathematics at Arizona

State University. His current research interests lie in

classical and quantum semiconductor device simulation,

computational fluid dynamics, astrophysical flows, and the

modelling and simulation of ion transport in the channels

of cellular membranes.

Anne Gelb is Associate Professor of Mathematics at

Arizona State University. Her main interests are the study

of higher order methods for solving partial differential

equations, in particular the applications of spectral

methods to discontinuous functions. A large part of her

research has been devoted to restoring the exponential

convergence qualities to spectral methods for discontinu-

ous functions. Recently she has become interested in

central difference schemes applied to conservation laws,

with applications to semiconductor device simulation and

astrophysical jets.

Justin Hernandez was a graduate student in Mathematics

at Arizona State University when this work was

completed. Currently he is working as a researcher in

computational mathematics.


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Journal ofEngineeringVolume 2014

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Civil EngineeringAdvances in

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The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

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